The backward shift on spaces of analytic functions

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In this project, the PI plans to investigate the `continuation' properties of the invariant subspaces of the backward shift operator on various spaces of analytic functions on the unit disk. In 1970, Douglas, Shapiro, and Shields showed that functions belonging to these invariant subspaces of the Hardy space posses special continuation properties to the exterior disk. In more recent investigations, this idea of `continuation' has been shown to be ubiquitous in that it appears to take place, in one form or another, in many other settings belong the Hardy space case. This proposal plans to get at the heart of the nature of these continuations and why they occur in the first place. This project falls under the broad heading of the field of mathematical analysis which, besides its beauty and elegance, makes many connections and has its roots in problems connected with physics and engineering. In fact, the concept of `continuation' properties of analytic functions has been recently studied by the engineer S. Darlington who connected these `continuations' to properties of electrical circuit design.